In addition to Shannon diversity, reported in the manuscript, we conducted the analysis using richness as our focal diversity metric. During the with seed rain stage, external seed addition is added to the simulated plant communities. This external seed addition is calculated to be 100% of the average monoculture seed rain across all species. Therefore, even rare species are able to maintain low population sizes. For this reason, there is often no slope within each planted species richness treatment. In other words, all communities that began with 32-species will continuously contain 32-species, because of seed addition. We thus report only the across-richness treatment models, omitting those within-richness treatments because they are invalid models.
Mirroring Shannon diversity in the manuscript, our models for the across-treatment effect were encoded as: Biomass ~ -1 + Stage + Stage:Richness. All models successfully converged, with Rhat values of 1.0, and posterior predictive checks (PPC) were used to visually validate the model fits.
The relationship between richness and total biomass was qualitatively similar to that of Shannon diversity. The direction and magnitude of the relationship between richness a total biomass are consistent across all of the models. The only variation between results is that in Forest2, the significance of the seed rain and no seed rain estimates both change; the seed rain slope becomes significant, though maintaining almost no slope, and further the no seed rain phase becomes insignificant, though again maintaining its general slope.
Again, the general structure of the relationship between the communities’ underlying coexistence dynamics and their emergent BEF relationship are maintained. The only qualitative difference is that the slope of the interaction is significantly less steep during the seed rain phase. This results stems from richness not fully capturing the changes in species composition within the communities. Because seed addition ensures most species are likely present within the plots, richness during the seed rain phase is unlikely to change.
This section of the document describes the statistical models’ validation, using richness as the focal biodiversity metric and total biomass as the focal ecosystem function.
Important terms:
Stage: With seed rain, without seed rainNinitial: Planted species richnessClark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 23.04 1.37 20.34 25.70 1.00 2573 2369
## StageWithoutseedrain 6.06 1.67 2.73 9.28 1.00 2327 2505
## StageWithseedrain:Richness 1.92 0.10 1.73 2.12 1.00 2368 2241
## StageWithoutseedrain:Richness 7.03 0.31 6.45 7.62 1.00 2590 2342
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 18.61 0.48 17.72 19.56 1.00 3731 2812
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5464343 0.01616722 0.513239 0.5760942
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 60.23 0.96 58.33 62.13 1.00 2863 2971
## StageWithoutseedrain 60.29 0.96 58.43 62.16 1.00 2637 2930
## StageWithseedrain:Richness 0.66 0.06 0.53 0.78 1.00 2990 2743
## StageWithoutseedrain:Richness 0.87 0.06 0.75 1.00 1.00 2795 3117
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 13.85 0.36 13.20 14.56 1.00 3757 2652
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2838514 0.02308112 0.2382054 0.3282565
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 50.11 0.71 48.72 51.49 1.00 2202 2346
## StageWithoutseedrain 49.40 0.77 47.86 50.90 1.00 2538 2406
## StageWithseedrain:Richness 0.26 0.05 0.17 0.34 1.00 2521 2422
## StageWithoutseedrain:Richness 0.48 0.08 0.32 0.65 1.00 2584 2522
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.87 0.26 9.37 10.39 1.00 3770 2678
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.08364221 0.01798936 0.05061246 0.1215145
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 42.31 1.52 39.32 45.28 1.00 2525 2306
## StageWithoutseedrain 38.57 1.82 35.01 42.13 1.00 2707 2646
## StageWithseedrain:Richness 1.21 0.14 0.94 1.48 1.00 2691 2643
## StageWithoutseedrain:Richness 3.72 0.53 2.70 4.75 1.00 2497 2409
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 21.32 0.53 20.34 22.39 1.00 3861 2536
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1493185 0.02177845 0.1068745 0.1922317
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 30.87 0.73 29.44 32.31 1.00 2551 2759
## StageWithoutseedrain 23.49 0.91 21.69 25.24 1.00 2480 2764
## StageWithseedrain:Richness -0.00 0.05 -0.10 0.09 1.00 2550 2724
## StageWithoutseedrain:Richness -1.22 0.17 -1.56 -0.88 1.00 2355 2323
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 10.59 0.27 10.08 11.15 1.00 3515 2837
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2986958 0.02404684 0.2513005 0.3456881
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 76.08 0.64 74.83 77.37 1.00 2698 2493
## StageWithoutseedrain 80.36 0.91 78.56 82.11 1.00 2441 2587
## StageWithseedrain:Richness 0.04 0.04 -0.04 0.12 1.00 2851 2582
## StageWithoutseedrain:Richness -1.81 0.30 -2.39 -1.23 1.00 2383 2458
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.20 0.23 8.75 9.65 1.00 3151 2850
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.05185961 0.01469274 0.02567783 0.08352739
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.